Tag Archives: trigonometry

Sideband #81: Tangent Cones

It’s been a while since my last Sidebands post. That’s partly because I’ve been working on a project that I’m sure will become a multi-post series and thought it would be nice to start with #81. But I’m not done (or actually started on the writing) yet, and this one has also been lurking for a while.

Essentially, I needed to figure out how to join a cone to a sphere in a seamless way (as in the picture here). This requires the sides of the cone meet the sphere at a tangent point.

It’s yet another case of actually needing the trigonometry I learned in school.

Continue reading


BB #92: Actually Using Trig

Today (3/14) is Pi Day. People everywhere (or at least math geeks everywhere) are baking decorated pies (or cakes or cookies) to celebrate. And while this is yet another math-y post, it’s not about pi. I’m more of a tau guy, anyway, so I celebrate Tau Day (6/28), because I get twice the (pizza) pie.

Today is also Albert Einstein’s birthday, which I’ve always thought was a cool coincidence. He’s 145 now (and still being widely misquoted).

But this post isn’t about him either.

Continue reading


BB #83: The Born Rule is Pythagorean

It’s actually obvious and might fall under the “Duh!” heading for some, but it only recently sunk in on me that the Born Rule is really just another case of the Pythagorean theorem. The connection is in the way the coefficients of a quantum superposition, when squared, must sum to unity (one).

For that matter, Special Relativity, which is entirely geometric, is yet another example of the Pythagorean theorem, but that’s another story. (One I’ve already told. See: SR #X6: Moving at Light Speed)

The obvious connection is the geometry behind how a quantum state projects onto the basis eigenvectors axes.

Continue reading


Sideband #72: Trig Is Easy!

Trigonometry is infamously something most normal people fear and loath. Or at least don’t understand and don’t particularly want to deal with. (In fairness, it doesn’t pop up much in regular life.) As with matrix math, trig often remains opaque even for those who do have a basic grasp of other parts of math.

Excellent and thorough tutorials exist for those interested in digging into either topic, but (as with matrix math) I thought a high-altitude flyover might be helpful in pointing out important concepts.

The irony, as it turns out, is that trig is actually pretty easy!

Continue reading


Sideband #63: Matrix Rotation

For me, the star attraction of March Mathness is matrix rotation. It’s a new toy (um, tool) for me that’s exciting on two levels: Firstly, it answers key questions I’ve had about rotation, especially with regard to 4D (let alone 3D or easy peasy 2D). Secondly, I’ve never had a handle on matrix math, and thanks to an extraordinary YouTube channel, now I see it in a whole new light.

Literally (and I do mean “literally” literally), I will never look at a matrix the same way again. Knowing how to look at them changes everything. That they turned out to be exactly what I needed to understand rotation makes the whole thing kinda wondrous.

I’m going to try to provide an overview of what I learned and then point to a great set of YouTube videos if you want to learn, too.

Continue reading


Beautiful Math

Take a moment to gaze at Euler’s Identity:

Eulers Identity

It has been called “exquisite” and likened to a “Shakespearean sonnet.” It has earned the titles “the most famous” and “the most beautiful” formula in all of mathematics, and, in a mere seven symbols, symbolizes much of its foundation.

Today we’re going to graze on it!

Continue reading


Happy Tau Day!

tau-1This might seem like another math post… but it’s not! It’s a geometry post! And geometry is fun, beautiful and easy. After all, it’s just circles and lines and angles. Well, mostly. Like anything, if you really want to get into it, then things can get complex (math pun; sorry). But considering it was invented thousands of years ago, can it really be that much harder than, say, the latest smart phone?

Even the dreaded trigonometry is fairly simple once you grasp the basic idea that the angles of a triangle are directly related to the length of its sides. (Okay, admittedly, that’s a bit of a simplification. The (other two) angles of a right-angle triangle are directly related to the ratios of the length of its sides, but still.)

However, this isn’t about trig; this is about tau!

Continue reading